Expectation of a product of Brownian Motions

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SUMMARY

The expectation of the product of three standard Brownian motions, denoted as E[Bt1.Bt2.Bt3], can be evaluated using properties of independence and the definition of Brownian motion. If Bt1, Bt2, and Bt3 are independent and refer to non-overlapping intervals, the expectation can be simplified to E[Bt1]E[Bt2]E[Bt3]. Given that each Brownian motion has a mean of zero, the result is E[Bt1.Bt2.Bt3] = 0. If the intervals overlap, the analysis requires decomposition into non-overlapping processes.

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Let Bt1, Bt2 and Bt3 be standard Brownian motions with ~N(0,1).

Then what is E[Bt1.Bt2.Bt3] ?

Any help would be much appreciated.
 
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jamesa00789 said:
Let Bt1, Bt2 and Bt3 be standard Brownian motions with ~N(0,1).

Then what is E[Bt1.Bt2.Bt3] ?

Any help would be much appreciated.

Hey jamesa00789 and welcome to the forums.

What are the conditions for each BM? Are they independent? Do they refer to different intervals for the same process? Maybe some overlap in intervals?

If they are truly independent you can use the property that E[XY] = E[X]E[Y] and take it from there.
 
Yes they are of the same standard brownian motion at different time intervals.
 
jamesa00789 said:
Yes they are of the same standard brownian motion at different time intervals.

If they are are at non-overlapping intervals, then use the definition of the Brownian motion. If they are over-lapping, then decompose it into processes that are non-overlapping and take care of parts that are overlapping.

Using this, the fact that E[XY] = E[X]E[Y], and the definition of BM, what do you get?
 

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